Biggest difference I see between NH and "cowboys" is: the old-fashioned cowboy follows tradition (noting that "buck and spur" is not being everyone's idea of tradition; but I've seen training like that, and successfully too) and NH follows the horse's nature. So while both can be effective, the NH trainer has more options, since he adapts his methods to each particular horse. Of course he can judge wrongly; and cowboys aren't totally insensitive; but just in general, that's how I see it.
(I'm afraid I had to laugh looking at that young man's expression..."The worst that can happen. . .")